↑

Properties

Label 24.0.15693807744...5296.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{72}\cdot 7^{16}$
Root discriminant $29.27$
Ramified primes $2, 7$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, 0, 0, 650, 0, 0, 0, 0, 0, 0, 0, 117, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 117*x^16 + 650*x^8 + 1)
 
gp: K = bnfinit(x^24 + 117*x^16 + 650*x^8 + 1, 1)
 

Normalized defining polynomial

\( x^{24} + 117 x^{16} + 650 x^{8} + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 12]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(156938077449417789520626992646455296=2^{72}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.27$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(112=2^{4}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{112}(1,·)$, $\chi_{112}(67,·)$, $\chi_{112}(65,·)$, $\chi_{112}(9,·)$, $\chi_{112}(11,·)$, $\chi_{112}(15,·)$, $\chi_{112}(81,·)$, $\chi_{112}(107,·)$, $\chi_{112}(85,·)$, $\chi_{112}(23,·)$, $\chi_{112}(25,·)$, $\chi_{112}(71,·)$, $\chi_{112}(79,·)$, $\chi_{112}(29,·)$, $\chi_{112}(95,·)$, $\chi_{112}(99,·)$, $\chi_{112}(37,·)$, $\chi_{112}(39,·)$, $\chi_{112}(43,·)$, $\chi_{112}(109,·)$, $\chi_{112}(93,·)$, $\chi_{112}(51,·)$, $\chi_{112}(53,·)$, $\chi_{112}(57,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{9773} a^{16} + \frac{1154}{9773} a^{8} - \frac{4731}{9773}$, $\frac{1}{9773} a^{17} + \frac{1154}{9773} a^{9} - \frac{4731}{9773} a$, $\frac{1}{9773} a^{18} + \frac{1154}{9773} a^{10} - \frac{4731}{9773} a^{2}$, $\frac{1}{9773} a^{19} + \frac{1154}{9773} a^{11} - \frac{4731}{9773} a^{3}$, $\frac{1}{9773} a^{20} + \frac{1154}{9773} a^{12} - \frac{4731}{9773} a^{4}$, $\frac{1}{9773} a^{21} + \frac{1154}{9773} a^{13} - \frac{4731}{9773} a^{5}$, $\frac{1}{9773} a^{22} + \frac{1154}{9773} a^{14} - \frac{4731}{9773} a^{6}$, $\frac{1}{9773} a^{23} + \frac{1154}{9773} a^{15} - \frac{4731}{9773} a^{7}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{47}{9773} a^{17} + \frac{5373}{9773} a^{9} + \frac{21968}{9773} a \) (order $16$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 39019312.21180779 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{12}$ (as 24T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 24
The 24 conjugacy class representatives for $C_2\times C_{12}$
Character table for $C_2\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{8})\), 4.0.2048.2, \(\Q(\zeta_{16})^+\), 6.0.153664.1, 6.6.1229312.1, 6.0.1229312.1, \(\Q(\zeta_{16})\), 12.0.96717311574016.1, 12.0.49519263525896192.1, 12.12.49519263525896192.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$